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That subject might not be quite accurate, but let me clarify.

At discrete times t=1,2,..., with probability 1 events of type X and Y produced by independent random processes happen infinitely often, but the expected gaps between any two Xs or Ys is nonfinite. Is it true, and if so how might one prove, that with probability 1 some X and Y will occur simultaneously?

(I'm wondering if the proof that one-dimensional simple random walks infinitely often return can transfer to two dimensions by some general principle.)

Thanks!

Asaf Karagila
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Murcy Me
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While it is true that a simple two-dimensional random walk returns to the start with probability $1$ and the expected number of returns is infinite, despite the infinite expected return time for a simple one-dimensional random walk, this is not the case for higher dimensional random walks. See the earlier question Proving that 1- and 2-d simple symmetric random walks return to the origin with probability 1 for discussion and proofs

So if $X$ and $Y$ are the events of a return to the start of independent two-dimensional random walks, then probability of that they ever both occur simultaneously is like the return of a four-dimensional random walk, and so less than $1$

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Henry
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  • Thank you. I am not sure you exactly caught what I was asking, but you definitely answered it anyway! What I am taking away is that, if there were some simple argument from infinite recurrence of 1-d walks to probability one of recurrence in 2-d walks it would apply as well in higher dimensions. I think that's right. – Murcy Me Apr 30 '16 at 03:06
  • Yes: if there were an argument going from 1D to 2D random walks, then you might be able to use the same argument to go from 2D to 4D, but in fact you cannot. – Henry Apr 30 '16 at 09:03